The Number of Permutations with Exactlyr132-Subsequences IsP-Recursive in the Size!

Abstract

AbstractProving a first nontrivial instance of a conjecture of Noonan and Zeilberger we show that the numberSr(n) of permutations of lengthncontaining exactlyrsubsequences of type 132 is aP-recursive function ofn. We show that this remains true even if we impose some restrictions on the permutations. We also show the stronger statement that the ordinary generating functionGr(x) ofSr(n) is algebraic, in fact, it is rational in the variablesxand1−4x. We use this information to show that the degree of the polynomial recursion satisfied bySr(n) isr